EconPapers    
Economics at your fingertips  
 

Conjugate Direction Methods and Polarity for Quadratic Hypersurfaces

Giovanni Fasano () and Raffaele Pesenti ()

Journal of Optimization Theory and Applications, 2017, vol. 175, issue 3, No 8, 764-794

Abstract: Abstract We use some results from polarity theory to recast several geometric properties of Conjugate Gradient-based methods, for the solution of nonsingular symmetric linear systems. This approach allows us to pursue three main theoretical objectives. First, we can provide a novel geometric perspective on the generation of conjugate directions, in the context of positive definite systems. Second, we can extend the above geometric perspective to treat the generation of conjugate directions for handling indefinite linear systems. Third, by exploiting the geometric insight suggested by polarity theory, we can easily study the possible degeneracy (pivot breakdown) of Conjugate Gradient-based methods on indefinite linear systems. In particular, we prove that the degeneracy of the standard Conjugate Gradient on nonsingular indefinite linear systems can occur only once in the execution of the Conjugate Gradient.

Keywords: Polarity in homogeneous coordinates; Quadratic hypersurfaces; Conjugate Gradient method; Indefinite linear systems; 90C30; 65K99; 51N15 (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://link.springer.com/10.1007/s10957-017-1180-6 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:joptap:v:175:y:2017:i:3:d:10.1007_s10957-017-1180-6

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10957/PS2

DOI: 10.1007/s10957-017-1180-6

Access Statistics for this article

Journal of Optimization Theory and Applications is currently edited by Franco Giannessi and David G. Hull

More articles in Journal of Optimization Theory and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().

 
Page updated 2025-03-20
Handle: RePEc:spr:joptap:v:175:y:2017:i:3:d:10.1007_s10957-017-1180-6